If you can read Italian, here you can find full biographic details of his life. What follows is a short summary.
As was customary in Italy (and not only in Italy) until at least the 18th century, his name was written in many different ways, like Urbano (Giovan Francesco) Davisi, or D'Aviso, Avvisi, De Aviso, De Avisus and in the Latinized form Avisius, and also under the anagram "Buonardo Savi" used for Galileo's Trattato della sfera.
He was born in Rome on 25 May 1618, the fifth son of Giovanni Andrea, know as "berrettaro" (beret maker).
He entered the Jesuati Order (not the Jesuits) in 1636, where he studied philosophy and theology. From Rome he moved to Bologna to study mathematics under Cavalieri (also a Jesuat).
In 1650 he took care of the reprinting of Cavalieri's Specchio Ustorio, "to satisfy the intentions" of the master (died three years before, in 1647).
He returned to Rome in 1650 and in 1656 became General Procurator and Prior of the convent of Saints John and Paul. In 1656, under the name of "Buonardo Savi", he printed the Galileo's manuscript of the Trattato della Sfera together with some astronomical teachings "taught by Cavalieri to his pupils".
The two letters (Due lettere scritti dal rev. padre fra Urbano Davisi..., Bologna 1667) can be considered a short treatise on meteorology. In the first, to give reasons for various meteorological effects (rain, snow, ...), he places the element of fire at the center of the earth and not in the concave of the Moon as Aristotle wanted. The subject of the letter to Geminiano Montanari is another classic question of meteorology: the origin of the sources and rivers.
He was interested in applied hydraulics and, among other things, in the navigation of the Tiber, as evidenced by the title of one of his manuscripts (Tracratus de Tyberis navigatione ...), now lost.
In 1682, another Trattato della Sfera (different the Galileo's one) came out. Much has been discussed, but without reaching a definitive conclusion, on the authorship of this work, since in 1690, after its dead, it was reprinted under the modified title Sfera astronomica, which indicated as author Cavalieri and relegated Davisi to the role of simple publisher.
In reality, the treatise is largely by Cavalieri: divided into two parts - one setting out the doctrine of the sphere, the second made up of astronomical practices (many of which Davisi had already published in 1656) - and preceded by a life of Cavalieri, it is a partial translation, with additions and changes to Sphaera seu doctrinae sphaericae tractatus... authore F. Bonaventura Cavalerio, dated 1642 and preserved in the Library of the University of Bologna. It is still unclear why Davisi, always ready to acknowledge his debts to the master (I think this is why Cioffarelli speaks about "the awkward attempt to attribute to himself the edited version of the work"), did not do so in this case as well. Nor it is possible to establish whether the "reprint" of the 1690 is the result of an initiative of the bookseller or it represents the fulfilment of a will of Davisi.
Davisi died after a long and painful infirmity on 17th September 1686; he was buried in the church of S. Giovanni della Malva.
Here the two different Trattato della Sfera:
Trattato della sfera di Galileo Galilei, Roma 1656. The authors are Galileo and "Buonardo Savi": the first part is taken from lessons given by Galileo, the second part ("Prattiche Astronomiche") are based on the astronomical teachings of Cavalieri with Davisi's additions.
Sfera Astronomica del P. Bonaventura Cavalieri, Roma 1690. The work is preceded by a life of Cavalieri, the first of its kind and the principal document for the mathematician's biography, definitely due to Davisi.
The construction of the pentagon and of the hexagon at page 255 is the following:
Con l’occasione di questo disegnare le figure, ti voglio dare il modo di descrivere, e formare mechanicamente un Pentagono, che è una delle più difficili figure da disegnare, e pure è la più facile, che si facci in natura, perché non è altro, che un semplice nodo. Prenderai per tanto una striscia di carta della larghezza, che tu vorrai, e che habbi li lati paralleli, e con quella procura di fare un nodo, come se fosse una corda, auertendo però che la carta resti sempre stesa nelle piegature, che stringendola tanto che resti ben tirata, se taglierai con le forbici li capi che auanzano, hauerai fatto un Pentagono giustissimo.
Farai anco la figura Esagona se prenderai due striscie di carta di eguale lunghezza, e con li lati paralleli, e procurerai di fare con esse un nodo, facendo che le punte dell’incuruatura, che hauerai fatta di una striscia, passino per l’aperto dell’incuruatura dell’altra, che stringendole adattamente, e che mantenghino sempre la loro larghezza, tagliando l’auanzi delle punte, hauerai fatto un Esagono perfettissimo.
As we are drawing some figures, I want to give you the way to describe, and mechanically form, a Pentagon, which is one of the most difficult figures to draw, and also the easiest that there is in nature, because it is nothing more than a simple knot. You will therefore take a strip of paper of the width that you want, and that has parallel sides, and with it you will make a knot, as if it were a rope, but being careful that the paper remains always stretched during the folding, and tightening it so that it remains well pulled, then, if you cut with scissors the ends that advance, you will have made a perfect Pentagon.
You will also make a figure in the shape of a hexagon if you take two strips of paper of equal length, and with the sides parallel, and make a knot with them, making the tips of the curvature of one strip, pass through the opening of the curvature of the other, so that by tightening them properly, always keeping their width and cutting the leftovers of the tips, you will have made a perfect Hexagon.